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Everywhere I look I see that the period of a state $i$ in a Markov chain is given by $$ \gcd\{n>0 : P_{ii}^n>0 \} $$ but what do we mean if the set $\{n>0 : P_{ii}^n>0 \} = \emptyset$? For instance in a two state Markov chain with $P_{00}=1$ and $P_{10}=1$, what is the period of the state $i=1$?

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Not many authors pay attention to this case, it is true. One author who does is Pierre Bremaud. In his Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues he explicitly defines the period to be "infinity" when return is impossible. I guess that definition is as reasonable as any.

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Periods generally only make sense in irreducible components of a Markov chain. In your example, 0 and 1 are not in the same component. Communication is a two way street, you need to be able to get from $i$ to $j$ and from $j$ to $i$ with positive probability.

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What does that have to do with anything? Period is a state property not a transition property. We speak of the period of state $i$, not of the period between $i$ and $j$. –  nullUser Feb 27 '13 at 2:12
    
@nullUser: Period is precisely a "transition property." $P^n_{ii}$ is precisely the $n$ state return probability for a state $i$ which is a sum over every possible intermediate state. How else do you plan to come back to a state $i$ if not through some state $j$ which it communicates with? –  Alex R. Feb 27 '13 at 3:18

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